pdf-file - Institut für Theoretische Physik
pdf-file - Institut für Theoretische Physik
pdf-file - Institut für Theoretische Physik
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⎛<br />
⎜<br />
⎝ −ΠD,eff t,i<br />
<br />
HD × n <br />
⎞ ⎛<br />
⎟ ⎜<br />
⎠ = ⎝<br />
i<br />
ηs<br />
⎞ ⎛<br />
ζs ⎟ ⎜<br />
⎠ ⎝<br />
¯ζs αs<br />
∆vt,i<br />
⎞<br />
⎟<br />
⎠<br />
c ∆Et,i<br />
(100)<br />
where −Π D,eff<br />
t,i<br />
≡−Π D t,i + H D t,i Bn + E D t,i Dn + 1<br />
4 σsf∆Et,j, and<br />
¯γs = −γs , ¯ ζs = −ζs . (101)<br />
αs,βs,ηs,κs,σs > 0 . (102)<br />
These are altogether 21 boundary conditions. They suffice to determine (i) all<br />
(outgoing) hydrodynamic modes of the dielectric medium, 9 for each side; (ii)<br />
the normal components of B + B D and D + D D , (iii) the lab velocity. See [14]<br />
for a more detailed consideration (in single-component liquids).<br />
4.2 The Dielectric-Dielectric Interface, with Phase Transition<br />
Mass and concentration currents across the interface renders the consideration<br />
slightly more complicated.<br />
The 12 continuity conditions remain:<br />
∆(Qn + Q D n )=0, (103)<br />
∆(ρvn − j D n )=0, (104)<br />
∆(ρc vn − j D c,n) = 0 (105)<br />
∆(Πnn − Π D nn) =Psf , (106)<br />
∆(Πt,i − Π D t,i) t1,i = t1 ·∇αsf , (107)<br />
∆(Πt,i − Π D t,i) t2,i = t2 ·∇αsf<br />
(108)<br />
∆(B D n + Bn) =0, (109)<br />
∆(Et + E D t )=0, (110)<br />
∆(D D n + Dn) =−σsf , (111)<br />
∆(Ht + H D t )=n × jel,sf/c . (112)<br />
The rest of 9 boundary conditions must now be deduced from R sf . Combining<br />
Eq(103) and (104), we obtain<br />
0=∆(Qn + Q D n ) − (µ + c 2 )∆(ρvn − j D n )<br />
=∆(Qn + Q D n +(µ + c 2 ) j D n − (µ + c 2 ) ρvn)<br />
+〈ρvn − j D n 〉∆µ, (113)<br />
17